- we introduced the pointer_step rc in the perspective of proving

[helm.git] / matita / matita / contribs / lambda_delta / ground_2 / star.ma

diff --git a/matita/matita/contribs/lambda_delta/ground_2/star.ma b/matita/matita/contribs/lambda_delta/ground_2/star.ma
index c183111ddf429584bd0aca1b6548a9ae9ceee1cd..1e46a48c6d536e131a1d0bcf9ba7c3de414387a6 100644 (file)
--- a/matita/matita/contribs/lambda_delta/ground_2/star.ma
+++ b/matita/matita/contribs/lambda_delta/ground_2/star.ma
@@ -18,7 +18,14 @@ include "ground_2/notation.ma".
 
 (* PROPERTIES OF RELATIONS **************************************************)
 
-definition Decidable: Prop → Prop ≝ λR. R ∨ (R → False).
+definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥).
+
+definition Confluent: ∀A. ∀R: relation A. Prop ≝ λA,R.
+                      ∀a0,a1. R a0 a1 → ∀a2. R a0 a2 →
+                      ∃∃a. R a1 a & R a2 a.
+
+definition Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R.
+                       ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
 
 definition confluent2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
                        ∀a0,a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
@@ -28,6 +35,10 @@ definition transitive2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
                         ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 →
                         ∃∃a. R2 a1 a & R1 a a2.
 
+definition bi_confluent:  ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
+                          ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 →
+                          ∃∃a,b. R a1 b1 a b & R a2 b2 a b.
+
 lemma TC_strip1: ∀A,R1,R2. confluent2 A R1 R2 →
                  ∀a0,a1. TC … R1 a0 a1 → ∀a2. R2 a0 a2 →
                  ∃∃a. R2 a1 a & TC … R1 a2 a.
@@ -99,10 +110,10 @@ lemma TC_transitive2: ∀A,R1,R2.
 qed.
 
 definition NF: ∀A. relation A → relation A → predicate A ≝
-   λA,R,S,a1. ∀a2. R a1 a2 → S a1 a2.
+   λA,R,S,a1. ∀a2. R a1 a2 → S a2 a1.
 
 inductive SN (A) (R,S:relation A): predicate A ≝
-| SN_intro: ∀a1. (∀a2. R a1 a2 → (S a1 a2 → False) → SN A R S a2) → SN A R S a1
+| SN_intro: ∀a1. (∀a2. R a1 a2 → (S a2 a1 → ⊥) → SN A R S a2) → SN A R S a1
 .
 
 lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
@@ -110,3 +121,38 @@ lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
 @SN_intro #a2 #HRa12 #HSa12
 elim (HSa12 ?) -HSa12 /2 width=1/
 qed.
+
+definition NF_sn: ∀A. relation A → relation A → predicate A ≝
+   λA,R,S,a2. ∀a1. R a1 a2 → S a2 a1.
+
+inductive SN_sn (A) (R,S:relation A): predicate A ≝
+| SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a2 a1 → ⊥) → SN_sn A R S a1) → SN_sn A R S a2
+.
+
+lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a.
+#A #R #S #a2 #Ha2
+@SN_sn_intro #a1 #HRa12 #HSa12
+elim (HSa12 ?) -HSa12 /2 width=1/
+qed.
+
+lemma bi_TC_strip: ∀A,B,R. bi_confluent A B R →
+                   ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. bi_TC … R a0 b0 a2 b2 →
+                   ∃∃a,b. bi_TC … R a1 b1 a b & R a2 b2 a b.
+#A #B #R #HR #a0 #a1 #b0 #b1 #H01 #a2 #b2 #H elim H -a2 -b2
+[ #a2 #b2 #H02
+  elim (HR … H01 … H02) -HR -a0 -b0 /3 width=4/
+| #a2 #b2 #a3 #b3 #_ #H23 * #a #b #H1 #H2
+  elim (HR … H23 … H2) -HR -a0 -b0 -a2 -b2 /3 width=4/
+]
+qed.
+
+lemma bi_TC_confluent: ∀A,B,R. bi_confluent A B R →
+                       bi_confluent A B (bi_TC … R).
+#A #B #R #HR #a0 #a1 #b0 #b1 #H elim H -a1 -b1
+[ #a1 #b1 #H01 #a2 #b2 #H02
+  elim (bi_TC_strip … HR … H01 … H02) -a0 -b0 /3 width=4/
+| #a1 #b1 #a3 #b3 #_ #H13 #IH #a2 #b2 #H02
+  elim (IH … H02) -a0 -b0 #a0 #b0 #H10 #H20
+  elim (bi_TC_strip … HR … H13 … H10) -a1 -b1 /3 width=7/
+]
+qed.